Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 145 / 2 / f

Table of contents
- Solution f

Solution

A proper colouring using exactly

k

colours first partitions the vertices into

k

nonempty independent colour classes and then injectively assigns

k

of the

x

named colours to those classes. These choices number

{n k}_{G} x^{\underline{k}} .

Summing over

k

counts every proper colouring exactly once; the expression is the chromatic polynomial

χ_{G} (x)

.

Solved by gpt-5.6-sol high.

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